Reform mathematics

Standards-based mathematics is one name for a reform method of mathematics instruction, usually based on recommendations published in 1989 by the National Council of Teachers of Mathematics (NCTM).[8] The original document was Principles and Standards for School Mathematics. It attempted to set forth a North American vision for precollege mathematics education in the United States and Canada. The recommendations were largely adopted by most education agencies, from local to federal levels, by the mid 2000s. Today, it still serves as the primary basis for many states' mathematics standards, for many federally funded textbook projects, and as an influence on standards in other nations.

Standards

A major goal of standards-based education is to ensure that "all children will succeed", by setting uniform standards of "what every child is expected to know and be able to do". By holding all participants in the system accountable through standards and assessments, this movement intends to have all students master the mathematics skill-set adopted by the educational agency.

2006 Focal Points

In 2006, NCTM issued a document called "Focal Points" which presented a more concise set of goals and objectives for each grade in elementary and middle schools. The "Focal Points" were perceived by the press (notably the Wall Street Journal (Sept 12, 2006), the New York Times, the Chicago Tribune and other newspapers) to be an admission that ambiguities in the previous standards had permitted the creation of curricula which had substantially reduced instruction in basic arithmetic facts. The 2006 document called for a strong emphasis of direct instruction of basic skills.

While the PSSM was championed by education theorists and administrators as raising standards for all students, it was sharply attacked by mathematicians, parents, and even some teachers over the new teaching methods which inspired lampooned exercises such as Mathland's Fantasy Lunch, Rainforest Algebra, and academic papers finding that teaching arithmetic harmed mathematical understanding. Some officials were quoted as valuing understanding processes more than learning one correct way to get one correct answer. Although still widely adopted in the United States and abroad by the mid-2000s, some education agencies reject the NCTM standards in favor of more traditional approaches such as Singapore Math and Saxon math.

In 2006, the NCTM released “Curriculum Focal Points,” a report urging that math teaching in kindergarten through eighth grade focus on a few basic skills, largely reversing the controversial stand taken in the landmark 1989 standards document which launched the math wars of the 1990s and 2000s. Francis Fennell, president of the council played down the degree of change the new report, and said that he resented talk of “math wars.” Interviews of many who were committed to the standards said that, like the 2000 standards, these merely refined and focused rather than renounced the original 1989 recommendations.

Nevertheless, many newspapers like the Chicago Sun Times reported that the "NCTM council has admitted, more or less, that it goofed". The new report cited "inconsistency in the grade placement of mathematics topics as well as in how they are defined and what students are expected to learn." The new recommendations are that students are to be taught the basics, including the fundamentals of geometry and algebra, and memorizing multiplication tables.

Many school districts and states are committed to curricula and frameworks based on the now-obsolete mathematics standards which many parents and citizens claim robbed their children of an education in basic arithmetic skills.

Terminology

Mathematics in this style have also been called "standards-based" instruction or "standards-based mathematics, or simply "reform mathematics".

Less favorable terminology which have appeared in press and web articles include fuzzy math, "Where's the math", "anti-math", "math for dummies", "no-math mathematics", rainforest algebra , "Math for women and minorities, and "new new math".

Traditional mathematics education has been called "Parrot Math" by critics. The direct instruction method has been criticized as "drill and kill".

Origins

Based on a consensus process that involved classroom teachers, mathematicians, and educational researchers from across the country, but later criticized by many people who actually used advanced mathematics for a living, the document sets forth a set of six principles (Equity, Curriculum, Teaching, Learning, Assessment, and Technology) that describe high-quality mathematics programs. Mathematical equity was a principle previously unseen in the field of mathematics, reflecting the influence of the politics of race, gender, and class of the 1960s and 1970s. Ten general strands or standards of mathematics content and processes were defined that cut across the school mathematics curriculum.

Specific expectations for student learning derived from beliefs of outcome-based education are described for ranges of grades (preschool to 2, 3 to 5, 6 to 8, and 9 to 12). The draft standards and the final standards make explicit goals that all students should learn higher level mathematics, particularly under-served groups such as minorities and women. These standards were made an integral part of nearly all outcome-based education and later standards-based education reform programs that were widely adopted by consensus across the United States by the 2000s.

Previously, de-facto standards had been set by textbook publishers. Mathematics texts were largely devoid of goals such as social justice and race and gender issues (equity). The new mathematics would reflect thinking in education since the activism of the 1960s and 1970s. Across education came the new context of the rise of multiculturalism and affirmative action as the primary goals of education rather than just academic content. This document built on several earlier standards documents produced by the National Council of Teachers of Mathematics [9] — including the Curriculum and Evaluation Standards for School Mathematics (1989), the Professional Standards for Teaching Mathematics (1991), and the Assessment Standards for School Mathematics (1995).

Dumbing down or raising the bar?

Critics decried the dumbing "down" of mathematics, and called for giving minorities the same standards and instruction which had served previous generations of mathematics and engineering professionals. Reformers pointed to the "basics" as being the dumbed down alternative to teaching representation, relating and communicating higher order thinking skills. The standards introduced new terminology such as mathematical power, which should be given to all students, not merely the successful few who were tracked into technical college majors, and number sense, which would go far beyond memorizing a few traditional computing methods.

As parents and math / science professionals revolted against curricula which in the case of Mathland and Investigations in Number, Data, and Space dispensed with instruction of traditional arithmetic as obsoleted by calculators in favor of writing, cutting, pasting, singing and coloring, the New York Times and Wall Street Journals made "Math Wars" a new headline story. While the standards were widely and nearly universally adopted by the mid-2000s, at the same time many schools, school districts and even states such as California effectively rejected the standards, instead adopting rigorous traditional content and skill based standards and supplementing or replacing standards based curricula with Saxon math and Singapore Math. Even the 2006 revision to NCTM guidelines lauded Singapore Math, though they would downplay headlines that that the standards had retreated back towards basic skills.

Emphasis on mathematical thinking

Traditional mathematics focuses on teaching algorithms that will lead to the correct answer. Because of this focus on rote application of algorithms, the traditional math student must always use the specific method that is being taught in today's lesson to get full credit for the correct answer. For example, in a traditional mathematics lesson on multiplication, a student who multiplies two numbers using the "cake layer" method is marked correct, but the student who arrived at exactly the same answer using the algebraic FOIL rule is marked incorrect. Next month, when the FOIL method is being taught, the student who uses the cake layer method will be marked incorrect. This kind of algorithmic dependence is de-emphasized in the NCTM standards.

• The NCTM recommends "decreased attention" for "finding exact forms of answers" in upper grades. (5.8.O) The presence of occasional minor errors is deemed less important than the overall thought process. When a complex equation has been solved to a level of simplicity, the final calculation may be relatively unimportant. For example, when solving a complex algebra problem, a student might stop when the remaining steps are simple arithemetic. The NCTM approach may require more attention from the teacher during grading.

• "Although written tests structured around a single correct answer can be reliable measures of performance, they offer little evidence of the kinds of thinking and understanding advocated in the Curriculum Standards." (EVAL.2)

• "Students might [...] believe that problem solving is always finding one correct answer using the right way. These beliefs, in turn, influence their actions when they are faced with solving a problem. Although such students have a positive attitude toward mathematics, they are not exhibiting essential aspects of what we have termed mathematical disposition." (EVAL.2)

The "decreased attention" statement above is one of many reasons for a bitter conflict between the self-described "traditionalists" and the reformers. The reformers do not oppose correct answers, but prefer to focus students' attention on the process leading to the answer, rather than the answer itself. In a formal evaluation, it is difficult to credit the work of a student who approaches a problem informally and exhibits substantially correct analysis while failing to use, for example, proper terminology or exact computations. The PSSM-supported approach would be to encourage the student to develop his or her arguments and to formalize them in appropriate mathematical language. In more traditional instruction, such student's answer would often be simply marked incorrect or insufficient.

It is important to note that PSSM offers only guidelines, along with some exemplary practices in supporting materials. However, the practical implementations of the guidelines sometimes fail in achieving the balance. The programs created in response to the reform have been criticized by all parties for overreaching in decreasing attention to some topics to the point of failing to teach them. In contrast, although the more traditional textbook series always claim adherence to and compliance with the standards, in practice, this amounts to lip service. The situation is complicated further by the fact that any textbook that hopes to be sold nationally must comply not only with the national standards, but also with a host of state standards--often contradictory in both content coverage and pedagogical intent.

The emphasis on analysis rather than the answer is common in advanced professional education, but many question whether such thinking is appropriate for children. For example, legal problems are often presented without a possibility of a single correct answer to encourage deep analysis of the factors that might contribute to an answer. As a result, finding a single answer generally results in a lower grade on law school exams. Although this parallel is not perfect, many professionals in mathematics and in education continue to argue that this is a valid approach.

High school freshman calculus and elementary algebra

According to the 1989 standards:

• It is "essential that in grades 5-8, students explore algebraic concepts in an informal way." (5-8.9)

• In grades 9-12, the mathematics curriculum should include the informal exploration of calculus concept" (9-12.13)

Traditional content

K-12 math should no longer necessarily cover the same content that has been traditionally taught under the headings "arithmetic", "algebra", and "geometry". According to the NCTM Standards: Introduction:

• Calculators and computers have "changed the very nature of the problems important to mathematics and the methods mathematicians use to investigate them".

• "quantitative techniques have permeated almost all intellectual disciplines. However, the fundamental mathematical ideas needed in these areas are not necessarily those studied in the traditional algebra-geometry-precalculus-calculus sequence." For example, many students need to understand practical medical statistics and financial calculations, which are largely ignored in the traditional mathematics curriculum, and very few students need to be proficient with spherical geometry, which is heavily emphasized in advanced math classes under a traditional curriculum.

• "For many non-mathematicians, arithmetic operations, algebraic manipulations, and geometric terms and theorems constitute the elements of the discipline to be taught in grades K-12. This may reflect the mathematics they studied in school or college rather than a clear insight into the discipline itself."

Technology

The standards put emphasis on using computers and calculators to render much manual calculation, and therefore instruction of such methods obsolete.

• "Contrary to the fears of many, the availability of calculators and computers has expanded students' capability of performing calculations. There is no evidence to suggest that the availability of calculators makes students dependent on them for simple calculations." (Intro)

• "Calculators must be accepted at the K-4 level as valuable tools for learning mathematics." (K-4.O)

• "Calculators enable children to compute to solve problems beyond their paper-and-pencil skills." (K-4.8)

• "The calculator renders obsolete much of the complex paper-and-pencil proficiency traditionally emphasized in mathematics courses." (5-8.O)

• "By assigning computational algorithms to calculator or computer processing, this curriculum seeks not only to move students forward but to capture their interest." (9-12.O)

Math appreciation and culture

According to the introduction:

• The first goal for all students is "they learn to value mathematics". ( Intro) ** See Quotes

• "Students should have numerous and varied experiences related to the cultural, historical, and scientific evolution of mathematics so that they can appreciate the role of mathematics in the development of our contemporary society." (Intro)

Problem Solving

General Problem-Solving Skills is seen as a way to solve problems without requiring teaching or memorizing specific math knowledge, such as common denominators or using a formula to compute an average.

• "Problem solving should be the central focus of the mathematics curriculum" (K-4.1)

• "Mathematical problem solving, in its broadest sense, is nearly synonymous with doing mathematics." (9-12.1)

• "A vital component of problem-solving instruction is having children formulate problems themselves." (K-4.1)

• The problem-solving strategies identified by the NCTM for the K-4 level are "using manipulative materials, using trial and error, making an organized list or table, drawing a diagram, looking for a pattern, and acting out a problem." At the 5-8 level the NCTM adds "guess and check". (K-4.1) and (5-8.1)

Critics cite "trial and error" and other content-independent "problem solving" skills as inappropriate ways of applying the fundamental strategy of "discovery learning" to math.

Mathematical communications

• "The communication standard (Standard 2) calls for the integration of language arts as children write and discuss their experiences in mathematics." (K-4.4)

• "Students should be encouraged to explain their reasoning in their own words." (5-8.3)

• "In mathematics, just as with a building, all students can develop an understanding and appreciation of its underlying structure independent of a knowledge of the corresponding technical vocabulary and symbolism." (9-12.14)

Critics agree communication skills may be even more important than math skills, but they're not math. Some parents worry about when their children writing, cutting, pasting and drawing more than computing. Many standards based exercises such as drawing pie charts and finding patterns in data appear to be Powerpoint rather than mathematics skills.

Math Wars: Controversy

Main article: Math wars

The document and curricula based on it were later fiercely opposed by many parents and mathematics professionals, and rejected by many states and school districts. As a major component of standards-based education reform, the NCTM standards were to be the mathematics counter part to whole language, which was received with much opposition, science, and history curriculum standards largely aligned with constructivism. Parents opposing reform mathematics often complain about decreased focus on basic computation skills and confusion caused by the increased emphasis on exploration and explanation. One parent, speaking out on the controversy in the highly regarded Ridgewood, NJ school district, put it thus: "It's like math for English majors...It was never equations. It was patterns, drawing circles, writing down numbers and explaining what you did."[10]

Compared to traditional sequences, many innovators had determined that an emphasis on elementary arithmetic was inefficient and unproductive for many students in the age of calculators. At the same time, since only a few students advanced to topics such as algebra, statistics, and calculus, it became important to move such topics to all students in elementary school. Since some of these topics exceed the level of math instruction of many parents and even teachers, the change left many adults feeling inadequate and threatened.

The NCTM Standards have led to changes in mathematics textbooks. Perhaps because of the abruptness of these changes, debate over textbooks has sometimes been polarized. This debate is known as the "Math Wars."

"Reform" textbooks teach concepts which used to be reserved for advanced students in higher grades, while de-emphasizing procedural skills such as long division. Advanced math texts often require the use of graphing calculators which cost over US $100. Reform texts favor problem-solving in new contexts over template word problems with corresponding examples. Reform texts emphasize written and verbal communication, working in a group, connections between concepts, connections between representations, activities such as cutting, pasting, and in the case of "Investigations" singing that were once reserved for kindergarten. Some devote so much space in print on "contexts" that the Core-Plus Mathematics Project includes a separate index of contexts with topics ranging from the board-game Monopoly to Nike and rain forests.

The emphasis introducing so many topics so early has been criticized as a curriculum that is 'a mile wide and an inch deep'. [11] Some experts believe that many topics are introduced too early, though the 1989 standards call for bringing the introduction to algebra as early as elementary school, and calculus in early high school. Core-Plus introduces linear algebra and matrices, once taught in freshman college calculus, as early as junior high school in some districts. Teaching advanced mathematics to all students rather than only the students on the most advanced track may appear to promote equity, but may not be appropriate for students who have not even mastered basic arithmetic. Some integrated math texts have been criticized as covering too many topics in a haphazard sequence, while spending only brief time on topics such as solving linear equations which a traditional algebra class might devote months to deep understanding of a few important single topics.

By contrast, "traditional" textbooks emphasize procedural mathematics, such as arithmetic calculation. They provide step-by-step examples with skill exercises. Unlike texts which have been called Rainforest algebra, they have far fewer pages, and they devote little or no space to real-life contexts such as running shoe companies or geography. The entirety of the first page on matrices in Core-Plus is devoted to information on running shoe companies and their stores, and contains no content about what matrices are. Unlike the standards, texts adapted from Japan or Singapore include students and examples from only a single culture. They do not include historical figures to enhance cultural or gender identity diversity, make no reference to "mathematical power" and contain little or no content with regard to social justice or the equity sought by the standards. However, current traditional textbooks usually include some projects and exercises meant to address the NCTM Standards. Most of the parent and mathematics professional objections in the math wars have been in regard to the dearth or poor quality of mathematical content. In contrast, the lack of diversity, context, or equity laid out by the 1989 standards has mainly objected to by the administrators and officials who have promoted standards based mathematics, and have opposed adoption of more traditional texts such as Saxon math and Singapore math.

Innovative curricula

• Mathland asks 2nd graders to cut out and paste a Fantasy Lunch.
• Investigations in Numbers, Data, and Space does not contain instruction on any traditional computation methods, except to mention that they are to be discouraged. These include regrouping, the standard formulas for computing an average and volume, the standard notation for longhand division, and using "R" to indicate a remainder.

It has no student textbook. It uses 100 charts and skip counting, but not multiplication tables to teach multiplication. A study shows that a second grader who used his knowledge of the properties of negative numbers got more accurate results than another who used a traditional borrowing method. The second grade book includes sheet music to the song "happy birthday" which is meant to be sung in several languages although words to other languages are not provided. Decimal math is taught using colored pencils and 10,000 grid chart. Converting to a common denominator to add fractions is not taught.

• The introductory chapter for Matrices in the Core-Plus Mathematics Project spends a page explaining about Nike and the running shoe industry. It contains no information on how to solve any of the charting or data problems which follow. A matrix is used as a place to put a data table at the high school level. It has a separate "index of contexts" to non-mathematics topics such as "Monopoly", "Nike" or "global warming".

By contrast, math texts such as Singapore Math and Saxon math contain very little content or methods outside the field of traditional mathematics, and they make little use of sophisticated graphic calculators, or pictures of many diverse cultural or disabled groups.

Compared to international standards

American Institutes for Research (AIR) February 7, 2005: "Because topics are mapped out in such a general way, the NCTM requirements risk exposing students to unrealistically advanced mathematics content in the early grades"....Students are exposed to these complicated mathematical topics in kindergarten and first grade at the same time they are learning basic addition and subtraction. Singapore, in contrast, considers algebraic concepts to be advanced rather than introductory mathematics, so algebra is not introduced until the sixth grade. The specificity and logic in Singapore’s spiral approach offer a more effective, better sequenced framework for a mathematical curriculum." The California mathematics framework is modeled on Singaporean and Japanese frameworks. It is similar to the Singapore framework in that it is organized around a varying set of mathematical topics appropriate to the grades in which they are taught."

See also

• Education in the United States
• Mathematically Correct, which opposes the NCTM standards
• Prof David Klein (California State University Northridge), who opposes the NCTM standards

External links

• NCTM standards online 120-day free access, otherwise the public is required to pay to purchase or view the standards.
  • Log in for full access to Principles and Standards online
  • Original 1989 Curriculum and Evaluation Standards
  • 1991 Professional Standards
  • 1995 Assessment Standards